Efficient Capacity-Achieving Codes for General Repeat Channels.

Given a probability distribution D over the nonnegative integers, a D-repeat channel acts on an input symbol by repeating it a number of times distributed as D. For example, the binary deletion channel (D=Bernoulli) and the Poisson repeat channel (D=Poisson) are special cases. We say a D-repeat channel is square-integrable if D has finite first and second moments. In this paper, we construct explicit codes for all square-integrable D-repeat channels with rate arbitrarily close to the capacity, that are encodable and decodable in linear and quasi-linear time, respectively. We also consider possible extensions to the repeat channel model, and illustrate how our construction can be extended to an even broader class of channels capturing insertions, deletions, and substitutions.Our work offers an alternative, simplified, and more general construction to the recent work of Rubinstein [3], who attains similar results to ours in the cases of the deletion channel and the Poisson repeat channel. It also slightly improves the runtime and decoding failure probability of the polar codes constructions of Tal et al. [1] and of Pfister and Tal [2] for the deletion channel and certain insertion/deletion/substitution channels. Our techniques follow closely the approaches of Guruswami and Li [4] and Con and Shpilka [5]; what sets apart our work is that to obtain our result, we show that a capacity-achieving code for the channels in question can be assumed to have an "approximate balance" in the frequency of zeros and ones of all sufficiently long substrings of all codewords. This allows us to attain near-capacity-achieving codes in a general setting. We consider this "approximate balance" result to be of independent interest, as it can be cast in much greater generality than just repeat channels.A full version of this paper is available at https://arxiv.org/abs/2201.12746.